arXiv subjects

Moab Croft

Publications and source records attributed to Moab Croft.

3 recordsLinked to original sources

arXiv2026

The Scattering Algebra of Physical Space: Wigner-Covariance and Fields

Following previous work, the Algebra of Physical Space (APS) is used to explore Wigner-covariance and spin/helicity fields within the Constructive Standard Model (CSM) of Particle Physics. The spinor formalism of the APS is used to derive explicit Wigner-covariance of Lorentz spinors, and equivalencies with the CSM are demonstrated via the Scattering Algebra (SA). Constructive fields for spin-1/2 and spin-1 are given in the APS and the necessary maps for Wigner-covariance are proposed. The forms of these spin fields are equivalent to Pauli spinors, thereby serving as a new bridge between the CSM and the study of Quantum Information. It is further seen that spin-1/2 fields of the APS are equivalent to the spin-1/2 fields in the CSM, but the massless cases cannot yet be handled within the SA due to various complications. Similarly, while spin-1 fields exist inside the APS, their SA equivalents deviate from the originally proposed spin-1 fields of the CSM; so further work is needed to prove correspondence between spin-1 fields of the APS and the CSM. Sample Lagrangian densities of the CSM are analyzed using the methods herein, are given geometric interpretations, and are connected to traditional Pauli Theory. Finally, the hff (Higgs and two massive fermions) Lagrangian density is used to determine the first constructive scattering amplitude that is defined purely in terms of the APS. Concerningly, this leads to an anti-Hermitian action term, and this surprise is later confirmed using traditional CSM techniques. Throughout this paper, the illuminating power of Geometric Algebra is clear: Everything has a geometric interpretation, and results can be accomplished matrix-free as well as coordinate-free.

hep-ph↗

arXiv2026

The Scattering Algebra of Physical Space: Squared Massive Constructive Amplitudes

The Algebra of Physical Space (APS) is used to explore the Constructive Standard Model (CSM) of particle physics. Namely, this paper connects the spinor formalism of the APS to massive amplitudes in the CSM. A novel equivalency between traditional CSM and APS-CSM formalisms is introduced, called the Scattering Algebra (SA), with example calculations confirming the consistency of results between both frameworks. Through this all, two significant insights are revealed: The identification of traditional CSM spin spinors with Lorentz rotors in the APS, and the connection of the CSM to various formalisms through ray spinor structure. The CSM's results are replicated in massive cases, showcasing the power of the index-free, matrix-free, coordinate-free, geometric approach and paving the way for future research into massless cases, amplitude-construction, and Wigner little group methods within the APS.

hep-ph↗

arXiv2025

The Wigner Little Group for Photons Is a Projective Subalgebra

This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a point-based view to a mirror-based view is a modern movement that allows for a more intuitive representation of geometric and physical entities, with vectors and their higher-grade counterparts viewed as hyperplanes. This reinterpretation simplifies the implementation of homogeneous representations of geometric objects within the Spacetime Algebra and enables a relative view via projective geometry. Then, after utilizing the intrinsic properties of Geometric Algebra, the Wigner little group is seen to induce a projective geometric algebra as a subalgebra of the Spacetime Algebra. However, the dimension-agnostic nature of Geometric Algebra enables the generalization of induced subalgebras to (1+n)-dimensional Minkowski geometric algebras, termed little photon algebras. The lightlike transformations (translations) in these little photon algebras are seen to leave invariant the (pseudo)canonical electromagetic field bivector. Geometrically, this corresponds to Lorentz transformations that do not change the intersection of the spacelike polarization hyperplane with the lightlike wavevector hyperplane while simultaneously not affecting the lightlike wavevector hyperplane. This provides for a framework that unifies the analysis of symmetries and substructures of point-based Geometric Algebra with mirror-based Geometric Algebra.

math-ph↗